Surface area and volume: cuboids and cylinders

Why does a jigsaw puzzle come in a big flat box? The box looks thin, so it feels like the cheap way to pack a puzzle. It is not. A flat box that holds one litre can take almost twice the card of a cube holding the same litre, so somebody is paying extra for that flat shape, on purpose. Leave the flat jigsaw box open for now; the card sums ahead will settle why it costs more.

Surface area is the total area of all the faces of a solid. For a hollow box or package, it is the material you need to make it. Volume is the space a solid fills, counted in cubes. For a solid block, volume is the material you need. A closed gift box for a candle is a cuboid, a box shape with six rectangular faces. This one is one hundred and twenty millimetres long, eighty wide and fifty tall. Its faces come in three matching pairs: top and bottom, front and back, and the two ends. Each pair is two identical rectangles. Top and bottom: two times one hundred and twenty times eighty, which is nineteen thousand two hundred square millimetres. Front and back: two times one hundred and twenty times fifty, which is twelve thousand. Now the two ends, each eighty by fifty: how many square millimetres do they add? Eight thousand square millimetres. Each end is eighty by fifty, four thousand, and there are two of them. Add the three pairs: nineteen thousand two hundred, plus twelve thousand, plus eight thousand, makes thirty-nine thousand two hundred square millimetres. One square centimetre is one hundred square millimetres, so divide by one hundred. The gift box uses three hundred and ninety-two square centimetres of card. Count one face from each pair and you get half the right answer. Writing two times in front of every pair means no face goes missing. Now the two one-litre boxes. A litre is one thousand cubic centimetres, the space inside a cube ten centimetres along each edge. The cube box's card is six identical faces of ten times ten, which makes six hundred square centimetres. The flat box is twenty-five by twenty by two centimetres. Length times width times height, twenty-five times twenty times two, is also one thousand cubic centimetres. How many square centimetres of card does the flat box use, all three pairs added? One thousand one hundred and eighty. The lid and base are a thousand, the long sides a hundred, and the short ends eighty. Same litre, almost twice the card. The flat box pays for a big lid, which carries a big picture of the puzzle and stacks flat on a shop shelf.

Volume counts the cubes that fill a solid. For a cuboid, one layer of cubes covers the base, length times width, and the layers stack up as many times as it is tall. That makes volume length times width times height. A phone dock is made from a beech block one hundred and fifty millimetres long, seventy wide and forty tall, with a slot cut across it for the phone. First, the units. One centimetre is ten millimetres, so a one-centimetre cube is ten by ten by ten small millimetre cubes, one thousand of them. One cubic centimetre is one thousand cubic millimetres. For a volume, you cube the length factor, and ten cubed is one thousand. The safest route is to convert the lengths before multiplying: fifteen centimetres, seven centimetres and four centimetres. What is the beech block's volume, in cubic centimetres? Four hundred and twenty cubic centimetres. Fifteen times seven is one hundred and five, and one hundred and five times four is four hundred and twenty. In millimetres, one hundred and fifty times seventy times forty is four hundred and twenty thousand cubic millimetres. Divide by one thousand, and it is four hundred and twenty again. Divide by one hundred instead and you get four thousand two hundred, ten times too much. A cubic centimetre is the bigger unit, so the number must be a thousand times smaller. One examiner's report, on a volume question, says: This maths question provided some challenge, especially at the point at which unit conversion took place making the numbers manageable for candidates. The conversion is where those answers came unstuck. The fix is the one used for the dock: convert each length to centimetres before multiplying, so the volume arrives in cubic centimetres. The slot is seventy millimetres long, twelve wide and twenty-five deep. In centimetres, that is seven by one point two by two point five. How much beech is left in the finished dock, once the slot is cut? Three hundred and ninety-nine cubic centimetres. Volume of slot: seven times one point two times two point five is twenty-one cubic centimetres. Beech left: four hundred and twenty take away twenty-one is three hundred and ninety-nine. Naming each step in words, volume of block, volume of slot, beech left, means the slot never gets forgotten.

A cylinder is a circle stretched along a straight line, like a tin or a length of dowel. Slide its circular end along its length, and the end sweeps out the volume. Volume of a cylinder equals the area of the end times the length: pi r squared times h, where h is the height or length. A pen pot is cast in Jesmonite, a hard casting material mixed from a powder and a liquid resin. Its outside is a cylinder eighty millimetres across and one hundred tall. The hollow inside is a cylinder sixty-four millimetres across and ninety deep. Which goes into pi r squared for the outside: eight centimetres, or four? Four, the radius. Halve the diameter on its own line before it goes anywhere near the formula. So the outside has radius four centimetres and height ten. The hollow has radius three point two centimetres, half of six point four, and depth nine. Volume of outside: pi times four times four times ten is five hundred and two point six five cubic centimetres. Keep every digit. The task: find the volume of the hollow, then take it from the outside to get the Jesmonite, rounding only at the end. Pause it there and work it out for yourself. Two hundred and thirteen cubic centimetres of Jesmonite. Volume of hollow: pi times three point two times three point two times nine is two hundred and eighty-nine point five three. Jesmonite: five hundred and two point six five take away two hundred and eighty-nine point five three is two hundred and thirteen point one two, so two hundred and thirteen. Put the diameter in where the radius goes, and the outside alone becomes pi times eight times eight times ten, two thousand and eleven cubic centimetres. That is four times too much. A cork label wraps once round the outside of the pot, the full height. Unroll it, and it is a rectangle, as long as the circumference and as tall as the pot. What area of cork does the label need, in square centimetres? About two hundred and fifty-one square centimetres. The circumference is pi times eight, twenty-five point one three centimetres. Times the height of ten, that is two hundred and fifty-one point three, which rounds to two hundred and fifty-one. Check the unit from the sum. Two lengths multiplied give square units, for a surface. Three lengths give cubic units, for a volume. One examiner's report records this: It was encouraging to see more candidates showing a logical sequence to their work in how they laid their responses out, giving a note to explain what they were doing, such as volume of cuboid and volume of cylinder for example. This approach is to be encouraged as much as possible. That is the layout used for the pen pot: volume of outside, volume of hollow, Jesmonite left. The habit is to write the name of each step first, then its sum.

Now an oak cube, a change of units and a dowel, each answered after a pause.

A solid oak cube is five centimetres along each edge. What is its surface area? One hundred and fifty square centimetres. Six faces of five times five, twenty-five, make one hundred and fifty. Now a units one: how many cubic millimetres make two cubic centimetres? Two thousand. Each cubic centimetre holds one thousand cubic millimetres. Here's a dowel, radius one centimetre and twenty long. What is its volume? About sixty-two point eight cubic centimetres. Pi times one times one times twenty is twenty pi. One more: back to the oak cube. What is its volume, in cubic centimetres? One hundred and twenty-five, which is five times five times five.

Back to the two litre boxes. The flat jigsaw box spends almost twice the card of the cube to hold the same litre, and the maker decides the big picture is worth paying for.

If those four went your way, this one's yours: thumb it up and your rewatch list stays short. If not, keep it saved, because the costing video uses every one of these sums, so they come back round.

Next in the chapter: Calculating material quantities and costs, where areas and volumes turn into sheets, lengths and prices.

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